Abstract

We study the nonuniformly elliptic, nonlinear system Under growth and regularity conditions on the nonlinearities f and g, we obtain weak solutions in a subspace of the Sobolev space by applying a variant of the Mountain Pass Theorem.

1. Introduction

We study the nonuniformly elliptic, nonlinear system

where

System , There, under appropriate growth and regularity conditions on the functions and , the weak solutions are exactly the critical points of a functional defined on a Hilbert space of functions in . In the scalar case, the problem

with and , has been studied by Mihailescu and Radulescu [12]. In this situation, the authors overcome the lack of compactness of the problem by using the the Caffarelli-Kohn-Nirenberg inequality

In this paper, we consider which may be a nonuniformly elliptic system. We shall reduce to a uniformly elliptic system by using appropriate weighted Sobolev spaces. Then applying a variant of the Mountain pass theorem in [9], we prove the existence of weak solutions of system in a subspace of (,).

To prove our main results, we introduce the following some hypotheses:

There exists a function such that , for all ,

for all there exists a positive constant such that

for all ,

There exists a constant such that

for all ,

Let be the usual Sobolev space under the norm

Consider the subspace

Then is a Hilbert space with the norm

It is clear that

and the embeddings are continuous. moreover, the embedding is compact see [8]). we now introduce the space

endowed with the norm

1.1. Remark

Since for all we have with and

1.2. Proposition

The set is a Hilbert space with the inner product

for all

Proof. It suffices to check that any Cauchy sequences in converges to . Indeed, let be a Cauchy sequence in Then

and is bounded. Moreover, by Remark , is also a Cauchy sequence in . Hence the sequence converges to ; i.e,

It follows that converges to and converges to in Therefore converges to and converges to for almost everywhere . Applying Fatou's lemma we get

Hence Applying again Fatou's lemma

We conclude that converges to in

1.3. Definition

We say that is a weak solution of system if

for all

Our main result is stated as follows.

1.4. Theorem

Let and are satisfied, the system has at least one non-trivial weak solution in .

This theorem will be proved by using variational techniques based on a variant of the Mountain pass theorem in [9]. Let us define the functional given by

for

where

2. Existence of weak solutions

In general, due to the functional may be not belong to (in this work, we do not completely care whether the functional belongs to or not).This means that we cannot apply directly the Mountain pass theorem by Ambrosetti-Rabinowitz (see [4]), we recall the following concept of weakly continuous differentiability. Our approach is based on a weak version of the Mountain pass theorem by Duc (see [9]).

2.1. Definition

Let be a functional from a Banach space in to . We say that is weakly continuously differentiable on if and only if the following conditions are satisfied is continuous on . For any , there exists a linear map from into such that For any , the map , vi is continuous on .

We denote by the set of weakly continuously differentiable functionals on . It is clear that , where is the set of all continuously Frechet differentiable functionals on . The following proposition concerns the smoothness of the functional

2.2. Proposition

Under the assumptions of Theorem 1.4, the functional given by is weakly continuously differentiable on and

for all

Proof. By conditions -- and the embedding is continuous, it can be shown (cf. [[5], Theorem A.VI]) that the functional is well-defined and of class . Moreover, we have

for all

Next, we prove that is continuous on . Let be a sequence converging to in , where , ., Then

and is bounded. Observe further that

Similarly, we obtain

From the above inequalities, we obtain

Thus is continuous on . Next we prove that for all

Indeed, for any, any and we have

Since

Applying Lebesgue's Dominated convergence theorem we get

Similarly, we have

Combining - , we deduce that

Thus is weakly differentiable on .

Let be fixed. We now prove that the map is continuous on . Let be a sequence converging to in . We have

It follows by applying Cauchy's inequality that

Thus the map , i is continuous on and we conclude that functional is weakly continuously differentiable on . Finally, is weakly continuously differentiable on

2.3. Remark

From Proposition we observe that the weak solutions of system correspond to the critical points of the functional given by Thus our idea is to apply a variant of the Mountain pass theorem in [9] for obtaining non-trivial critical points of and thus they are also the non-trivial weak solutions of system

2.4. Proposition

The functional given by satisfies the Palais-Smale condition.

Proof Let be a sequence in H such that

First, we prove that is bounded in . We assume by contradiction that is not bounded in . Then there exists a subsequence of such that as . By assumption

it follows that

Letting since we deduce that which is a contradiction. Hence is bounded in .

Since is a Hilbert space and is bounded in , there exists a subsequence of weakly converging to in . Moreover, since the embedding is continuous, is weakly convergent to in . We shall prove that

On the other hand, using the continuous embeddings together with the interpolation inequality , it follows that

Since the embedding is compact we have as Hence as and is proved.

On the other hand, by and it follows

Hence, by the convex property of the functional we deduce that

Relations and imply

Finally, we prove that converges strongly to in . Indeed, we assume by contradiction that is not strongly convergent to in . Then there exist a constant and a subsequence of such that for any Hence

With the same arguments as in the proof of , and remark that the sequence converges weakly to in , we have

Hence letting , from ) and we infer that

Relations and imply which is a contradiction. Therefore, we conclude that converges strongly to in and satisfies the Palais -Smale condition on

To apply the Mountain pass theorem we shall prove the following proposition which shows that the functional has the Mountain pass geometry.

2.5. Proposition

There exist and such that for all with There exists such that and

Proof. From , it is easy to see that

where in view of . It follows from that

uniformly for

By using the embeddings , with simple calculations we infer from that for small enough This implies .

By , for each compact set there exists such that

Let having compact support, for large enough from we have

where , Then and imply

Proof of Theorem 1.4. It is clear that . Furthermore, the acceptable set

where is given in Proposition is not empty (it is easy to see that the function by Proposition. and Propositions - , all assumptions of the Mountain pass theorem introduced in [8] are satisfied. Therefore there exists such that

and for all is a weak solution of system . The solution is a non-trivial solution by The proof is complete.